Composite and Background Fields in Non-Abelian Gauge Models

Article Composite and Background Fields in Non-Abelian Gauge Models

Pavel Yu. Moshin 1,* and Alexander A. Reshetnyak 1,2,3

1 Department of Physics, National Research Tomsk State University, 634050 Tomsk, Russia; [email protected] 2 Center for Theoretical Physics, Tomsk State Pedagogical University, 634061 Tomsk, Russia 3 Institute of Strength Physics and Materials Science Siberian Branch of Russian Academy of Sciences, 634021 Tomsk, Russia * Correspondence: [email protected]

Received: 19 October 2020; Accepted: 25 November 2020; Published: 30 November 2020

Abstract: A joint introduction of composite and background fields into non-Abelian quantum gauge theories is suggested based on the symmetries of the generating functional of Green’s functions, with the systematic analysis focused on quantum Yang–Mills theories, including the properties of the generating functional of vertex Green’s functions (effective action). For the effective action in such theories, gauge dependence is found in terms of a nilpotent operator with composite and background fields, and on-shell independence from gauge fixing is established. The basic concept of a joint introduction of composite and background fields into non-Abelian gauge theories is extended to the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion, as well as to the Gribov–Zwanziger theory.

Keywords: composite ﬁelds; background ﬁelds; Yang–Mills theory; Volovich–Katanaev model; Gribov–Zwanziger model; background effective action; gauge-dependence

1. Introduction Composite [1,2] and background [3–5] fields are widely used in quantum gauge theories. The attention to composite fields (see [2] for an overview) stems from the fact that the effective action for composite fields suggested in [1] has been applied to quantum field models such as [6–8], including the Early Universe, Inflationary Universe, Standard Model and SUSY theories [9–13]. The study of BRST-invariant renormalizability in Yang–Mills theories, which includes N = 1 SUSY formulations [14–16], the functional renormalization group [17–21] and the Gribov horizon [22–24], appears to be promising within the concept of soft BRST symmetry breaking [25–28] and the local composite operator technique [29,30] as applied to arbitrary backgrounds [31]. In its turn, the background field method [3–5] presents the quantization of Yang–Mills theories using background gauges [5,32,33] in such a way that provides an invariance of the effective action under the gauge transformations of background fields and reproduces physical results with essential simplifications in the Feynman diagrams, thereby providing insight into diverse quantum properties of gauge theories [34–43]; for recent developments, see [44–48]. The present article is devoted to quantum non-Abelian gauge models with composite and background fields, whose consistent analysis isfocused on Yang–Mills theories quantized using the Faddeev–Popov method [49] in a combined presence of composite and background fields. A joint treatment of Yang–Mills fields Aµ with composite and background ones requires a systematic consideration of these ingredients as interrelated. We forward the symmetry principle as such a systematic concept. In fact, suppose that a generating functional Z (J, L) of Green’s functions with

Symmetry 2020, 12, 1985; doi:10.3390/sym12121985 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1985 2 of 26

A composite fields is given, depending on sources JA for the quantum fields φ , as well as on sources m Lm for the composite fields σ (φ). A question then naturally arises of how one can introduce some background fields Bµ in a way that leads to an extended functional Z (B, J, L) reflecting the possible symmetries of Z (J, L). Let us next suppose that a generating functional Z (B, J) of Green’s functions, also with certain symmetries, is given in the background field method, and then the question is how m some composite fields σ (φ, B) with sources Lm can be introduced for the resulting Z (B, J, L) to inherit the given symmetries. It turns out that these two approaches are equivalent in the following sense. According to the first approach, a generating functional Z (J, L) is given,

Z i h i Z (J, L) = dφ exp S (φ) + J φA + L σm (φ) , (1) h¯ FP A m corresponding to the Faddeev–Popov action SFP (φ) of a Yang–Mills theory with composite fields m σ (φ). A background field Bµ can then be introduced by localizing the inherent global symmetry of Z (J, L) under SU(N) transformations (rotations for JA and tensor transformations for Lm) in such a way that Z (B, J, L) defined as Z (B, J, L) = Z (J, L)| (2) ∂µ→Dµ(B) is invariant under local SU(N) transformations of the sources JA, Lm, accompanied by gauge transformations of the field Bµ with an associated covariant derivative Dµ (B), where the precise meaning of ∂µ → Dµ (B) in (2) is given by (19) of Section2. The original SFP (φ) becomes thereby modified to the Faddeev–Popov action of the background field method, SFP (φ, B), which is related to SFP (φ) by so-called background and quantum transformations of this method (see Section 2.2). According to the second approach, a generating functional Z (B, J) is constructed using the background field method for Yang–Mills theories,

Z i h i Z (B, J) = dφ exp S (φ, B) + J φA , (3) h¯ FP A which implies S (φ, B) = S (φ)| . FP FP ∂µ→Dµ(B) m Some composite ﬁelds σ (φ, B) with sources Lm can then be introduced on condition that the resulting generating functional

Z i h i Z (B, J, L) = dφ exp S (φ, B) + J φA + L σm (φ, B) (4) h¯ FP A m should inherit the symmetry of Z (B, J) under local SU(N) rotations of the sources JA accompanied by gauge transformations of the background field Bµ with the covariant derivative Dµ (B). This symmetry requirement for Z (B, J, L) is satisfied by a local SU(N) tensor transformation law imposed on σm (φ, B) m and is provided by Bµ entering the composite fields σ (φ, B) by means of the covariant derivative Dµ (B), which implies σm (φ, B) = σm (φ)| (5) ∂µ→Dµ(B) for certain σm (φ), and thereby we return to the first approach. In the main part of the article, we implement the first approach as a starting point of our systematic analysis, assuming the composite fields to be local, whereas in the remaining part we show how the first and second approaches can be extended beyond the given assumptions by considering the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion [50] and the Gribov–Zwanziger theory [23,24]. Two-dimensional models of gravity [50–61] and supergravity [62–65] are of interest in view of their close relation to string and superstring theory. Simple two-dimensional models also provide a deeper insight into classical and quantum properties of gravity in higher dimensions, while in some Symmetry 2020, 12, 1985 3 of 26 cases these models are exactly solvable at the classical level. One of the two-dimensional gravity models that has been widely discussed at the classical [66–70] and quantum [71–75] levels is the model [50] suggested in the context of bosonic string theory with dynamical torsion [76] in order to address some problems of string theory. Thus, it has been shown [76], using the path integral approach, that a string with dynamical torsion has no critical dimension. The model [50] presents the most general theory of two-dimensional R2-gravity with independent dynamical torsion leading to second-order equations of motion for the zweibein and Lorentz connection. The model also contains solutions with constant curvature and zero torsion, thereby incorporating some other two-dimensional gravity models [51,52,61] whose actions, as compared to that of [50], do not allow a purely geometric interpretation. Being quantized in the background field method, the model yields a gauge-invariant background effective action [75]. In quantum Yang–Mills theories using differential (e.g., Landau or Feynman) gauges, the non-Abelian nature of the gauge group features the Gribov ambiguity [22] implying a residual gauge-invariance due to Gribov copies, which are removed by means of a Gribov horizon [22] implemented in the Gribov–Zwanziger model [23,24] being a quantum Yang–Mills theory in Landau gauge and including an additive horizon functional in terms of a non-local composite field [77]. The issue of bringing the horizon functional to other gauges has been settled due to the concept of finite field-dependent BRST transformations [26,28,78–84], which allows one to present the horizon functional using different gauges in a way consistent with the gauge-independence of the path integral, based on the Gribov–Zwanziger recipe [23,24] and starting from a BRST-invariant Yang–Mills quantum action in Landau gauge. The horizon functional in covariant Rξ gauges has been given by [26,28,82,85] (see also [86,87] for a BRST-invariant horizon) and later extended to the Standard Model in [88,89]. The article is organized as follows. In Section2, a generating functional of Green’s functions with composite and background fields in Yang–Mills theories is introduced, as well as a generating functional of vertex Green’s functions (effective action). Analyzing the dependence of the generating functionals of Green’s functions upon a choice of gauge-fixing, we find a gauge variation of the effective action in terms of a nilpotent operator depending on the composite fields and determine the conditions of on-shell gauge-independence. Besides, the effective action Γeff (B, Σ), with a background field Bµ and a set of auxiliary tensor fields Σm associated with σm, is found to exhibit a local symmetry m under the gauge transformations of Bµ combined with the local SU(N) transformations of Σ . In Section3, we examine the Volovich–Katanaev model [ 50] quantized according to the background field method in [75]. As an extension of our second approach (3)–(5) beyond the Yang–Mills case, the quantized two-dimensional gravity [75] is modified by the presence of local composite fields, and the corresponding background effective action is found to be gauge-invariant in a way similar to the Yang–Mills case. In Section4, we modify the Gribov–Zwanziger model [ 23,24] by introducing a background field, which extends our first approach (1), (2) beyond the Yang–Mills case with local composite fields and thereby yields a gauge-invariant background effective action. Section5 is devoted to concluding remarks. We use DeWitt’s condensed notation [90]. The Grassmann parity and ghost number of a quantity F are denoted by e (F), gh (F), respectively. The supercommutator [F, G} of any quantities F, G with definite Grassmann parities is given by [F , G} = FG − (−1)e(F)e(G)GF. Unless specified by an arrow, derivatives with respect to fields and sources are regarded as left-hand ones.

2. Generating Functionals and Their Properties Let us examine a generating functional Z (J, L) corresponding to the Faddeev–Popov action SFP (φ) of a Yang–Mills theory with local composite ﬁelds,

Z i h i Z (J, L) = dφ exp S (φ) + J φA + L σm (φ) , (6) h¯ FP A m Symmetry 2020, 12, 1985 4 of 26

m where Lm are sources to the composite ﬁelds σ (φ),

1 σm (φ) = Λm φAn ... φA1 , (7) ∑ A1...An n=2 n!

A i α α α i and JA are sources to the fields φ = A , b , c¯ , c composed by gauge fields A , (anti)ghost fields c¯α, cα, and Nakanishi–Lautrup fields bα, with the following distribution of Grassmann parity and ghost number:

A A A m A m e(φ ) = (0, 0, 1, 1) , gh(φ ) = (0, 0, −1, 1) , e(JA, Lm) = e(φ , σ ), gh(JA, Lm) = −gh(φ , σ ).

The Faddeev–Popov action SFP (φ), ←− SFP (φ) = S0 (A) + Ψ (φ) s , is given in terms of a gauge-invariant classical action S0 (A), invariant, δξ S0 (A) = 0, i i α under inﬁnitesimal gauge transformations δξ A = Rα (A) ξ with a closed algebra of gauge i generators Rα (A), ←− i j i j γ i γ i i δ R (A) R (A) − R (A) Rα (A) = F R (A) , F = const, R ≡ R , α, j β β, j αβ γ αβ α, j α δAj ←− and a nilpotent Slavnov variation s applied to a gauge Fermion Ψ (φ), e (Ψ) = 1,

←− α ←− 2 SFP (φ) = S0 (A) + Ψ (φ) s , Ψ (φ) = c¯ χα (φ) , s = 0, (8) where A←− i α α α γ β φ s = Rα (A) c , 0, b , 1/2Fβγc c . For the explicit ﬁeld content

i = (x, p, µ), α = (x, p) , µ = 0, . . . , D − 1, p = 1, . . . , N2 − 1, φA = Ap|µ, bp, c¯p, cp , (Aµ, b, c¯, c) ≡ Tp Ap|µ, bp, c¯p, cp , [Tp, Tq] = f pqrTr,

←− the ﬁeld variations φA s have the form

←− p p p p ←− pq q p pqr q r Aµ, b, c¯, c s = Dµ (A) , c , 0, b, g/2 [c, c]+ , Aµ, b , c¯ , c s = Dµ (A) c , 0, b , g/2 f c c , where pq pq prq r Dµ (A) ≡ ∂µ + gAµ , Dµ (A) = δ ∂µ + g f Aµ , p The classical action S0 (A) has the form (in the adjoint representation with Hermitian T ), Z Z 1 D µν 1 D p p|µν p q 1 pq S (A) = d x Tr FµνF = d x F F , Tr (T T ) = δ , (9) 0 2g2 4 µν 2 p p p prs r s Fµν ≡ Dµ (A) , Dν (A) , Fµν = ∂µ Aν − ∂ν Aµ + g f Aµ Aν ,

α p and the gauge Fermion Ψ (φ) = c¯ χα (φ) with some gauge-ﬁxing functions χα (φ) = χ (φ (x)) reads Z Z Ψ (φ) = dDx c¯pχp (φ) = 2 dDx Tr [c¯χ (φ)] , χ (φ) = Tpχp (φ) . (10) Symmetry 2020, 12, 1985 5 of 26

The Faddeev–Popov action SFP (φ) in (8), (9) is invariant under two kinds of global A A←− transformations: BRST transformations [91–93], δλφ = φ s λ, with an anticommuting parameter λ, A U 0A A p e (λ) = 1, and SU(N) rotations (ﬁnite φ → φ and inﬁnitesimal δςφ ) with even parameters ς ,

U 0 −1 p p p Aµ, b, c¯, c → Aµ, b, c¯, c = U Aµ, b, c¯, c U , U = exp (−gT ς ) , ς = const, p p p p prq r r r r q δς Aµ, b , c¯ , c = g f Aµ, b , c¯ , c ς , or, in a tensor form, via the adjoint representation with a matrix Mpq (ς),

0 p p p p pq q q q q pq pq pqr r 2 Aµ, b , c¯ , c = M (ς) Aµ, b , c¯ , c , M (ς) = δ + g f ς + O ς .

←− U 0 The classical action S0 (A) in SFP (φ) = S0 (A) + Ψ (φ) s is invariant under Aµ → Aµ as a p V 0 particular case (ξ (x) = const) of invariance under the ﬁnite form Aµ → Aµ of gauge transformations

0 −1 −1 −1 0 −1 p p p p Aµ = VAµV + g V ∂µV , Dµ (A ) = VDµ (A) V , V = exp (−gT ξ ) , ξ = ξ (x) , (11)

←− U ←− while the invariance of Ψ (φ) s under φA → φ0A reﬂects the explicit form of s and the fact that the gauge functions χp (φ) are local and constructed from the ﬁelds φA, structure constants f pqr and derivatives ∂µ, for instance, in Landau and Feynman gauges,

p µ p p p µ p χL (φ) = ∂ Aµ , χF (φ) = b + ∂ Aµ , (12)

U so that, in particular, χp (φ) transform as SU(N) vectors, with Ψ (φ) being invariant under φA → φ0A,

χ φ0 = Uχ (φ) U−1, δχp (φ) = g f prqχr (φ) ςq

Due to the same reason, local composite ﬁelds σm (φ) constructed from the ﬁelds φA, pqr structure constants f and derivatives ∂µ,

m p1···pk|µ1···µl σ (φ) = σ (φ(x)) , m = (x, p1 ··· pk, µ1 ··· µl) ,

U in the path integral (6) for Z (J, L) transform under φA → φ0A as tensors with respect to the indices p1,..., pk,

σ0 p1···pk|µ1···µl = Mp1q1 ··· Mpkqk σq1···qk|µ1···µl ,

p ···p |µ ···µ psrsq p ···rs···p |µ ···µ q {p}rqˆ p ···rˆ···p |µ ···µ q δςσ 1 k 1 l = g ∑ f σ 1 k 1 l ς ≡ g f σ 1 k 1 l ς , (13) rs∈{r1,...,rk} which generalizes the vector transformation. As a result, the exponential in the path integral (6) for A U 0A Z (J, L) is invariant under φ → φ , along with some global transformations of the sources JA, Lm,

U ··· (J L ) → (J L )0 J = (J p J p Jp J p ) L = Lp1 pk A, m A, m , A (A)µ, (b), (c¯), (c) , m µ1···µl , (14) in a tensor and inﬁnitesimal form,

p|µ p p p 0 pq q|µ q q q 0 p1···pk p q p q q1···qk (J , J , J , J ) = M (J , J , J , J ), Lµ ···µ = M 1 1 ··· M k k Lµ ···µ , (A) (b) (c¯) (c) (A) (b) (c¯) (c) 1 l 1 l (15) ( p|µ p p p ) = prq( r|µ r r r ) q p1···pk = {p}rqˆ p1···rˆ···pk q δς J(A), J(b), J(c¯), J(c) g f J(A), J(b), J(c¯), J(c) ς , δς Lµ1···µl g f Lµ1···µl ς ,

A m which entails the invariance of the source term JAφ + Lmσ (φ). Symmetry 2020, 12, 1985 6 of 26

p p Let us introduce an additional ﬁeld Bµ = BµT having a gauge transformation as in (11),

V 0 −1 −1 −1 Bµ → Bµ = VBµV + g V ∂µV , (16) with the inherent property

0 −1 Dµ B = VDµ (B) V , Dµ (B) ≡ ∂µ + gBµ , (17) and subject the exponential in the path integral (6) to the modiﬁcation i i exp [S (φ) + Jφ + Lσ (φ)] ≡ exp [S (φ, B) + Jφ + Lσ (φ, B)] , (18) h FP h FP ¯ ∂µ→Dµ(B) ¯ where the replacement ∂µ → Dµ (B) reads as ∂µ → Dµ (B) : ∂µ, • → Dµ (B) , • ⇒ Dµ (A) , • → Dµ (A + B) , • , (19) so that Fµν (A) → Dµ (A + B) , Dν (A + B) = Fµν (A + B) . (20)

Due to the transformation property (17) of the derivative Dµ (B), the generating functional Z (B, J, L) modiﬁed by the ﬁeld Bµ according to (18), (19),

Z i h i Z (B, J, L) = dφ exp S (φ, B) + J φA + L σm (φ, B) , (21) h¯ FP A m is invariant under a set of local transformations,

p pq q δξ Bµ = Dµ (B) ξ , ( p|µ p p p ) = prq( r|µ r r r ) q δξ J(A), J(b), J(c¯), J(c) g f J(A), J(b), J(c¯), J(c) ξ , (22) p1···pk = {p}rqˆ p1···rˆ···pk q δξ Lµ1···µl g f Lµ1···µl ξ , given by the gauge transformations (16) of the ﬁeld Bµ combined with a localized form U (ς) → V (ξ) p of the transformations (14), (15) for the sources JA, Lm with inﬁnitesimal parameters ξ ,

V 0 Bµ, JA, Lm → Bµ, JA, Lm .

The invariance property Z (B0, J0, L0) = Z (B, J, L) can be established by applying to the transformed path integral Z (B0, J0, L0) a compensating change of the integration variables:

p p p p prq r r r r q V 0 δξ Aµ, b , c¯ , c = g f Aµ, b , c¯ , c ξ , Aµ, b, c¯, c → Aµ, b, c¯, c , whose Jacobian equals to unity in view of the complete antisymmetry of the structure constants. The invariance of Z (B, J, L) can be recast in the form ( −→ −→ Z h pq i δ p ···rˆ···p δ dDx D (B) ξq + gξq f {p}rqˆ L 1 k µ p µ1···µl p1···pk δBµ δLµ1···µl  −→ −→ −→ −→  δ δ δ δ  + q prq r|µ + r + r + r ( ) = gξ f J(A) | J(b) p J(c¯) p J(c) p  Z B, J, L 0. (23) p µ δJ δJ δJ  δJ(A) (b) (c¯) (c) Symmetry 2020, 12, 1985 7 of 26

2.1. Effective Action and Gauge Dependence Let us present a generating functional of vertex Green’s functions and examine its gauge- dependence properties. To do so, we ﬁrst introduce an extended generating functional Z (B, J, L, φ∗),

Z i h i Z (B, J, L, φ∗) = dφ exp S (φ, φ∗, B) + J φA + L σm (φ, B) , (24) h¯ ext A m

∗ with an extended quantum action Sext (φ, φ , B) deﬁned as

∗ ∗ A←− ←− ←− ←− ←− Sext (φ, φ , B) = S (φ, B) + φ (φ s q), s q = s , s q s q = 0, FP A ∂µ→Dµ(B)

∗ A←− where φA is a set of antiﬁelds introduced as sources to the variations φ s q. Given the properties ←− ←− ←− ∗ SFP (φ) s = 0 and s q s q = 0, the extended quantum action Sext (φ, φ , B) satisﬁes the identity

∗ ←− ←− ←− Sext (φ, φ , B) s q = S (φ, B) s q = S (φ) s = 0, FP FP ∂µ→Dµ(B) having an equivalent form, implied by the antisymmetry of the structure constants: −→ −→ A δ δ ( ) = ≡ (− )e(φ ) 2 = ∆ exp i/¯h Sext 0, ∆ 1 A ∗ , ∆ 0. (25) δφ δφA

∗ In view of (24), (25), the variation δΨZ (B, J, L, φ ) related to a change δΨ (φ, B) has the form

Z ∗ i h A m i i ∗ δ Z (B, J, L, φ ) = dφ exp J φ + Lmσ (φ, B) δ exp S (φ, φ , B) , (26) Ψ h¯ A Ψ h¯ ext i i ←− i i ←− ←− δ exp S = δΨ s exp S = −∆ δΨ exp S , δΨ s = δΨ (φA s ), Ψ h¯ ext h¯ q h¯ ext h¯ ext q ,A q which can be represented as −→ ! −→ −→ ! h¯ δ δ i h¯ δ δΨZ = δΨ,A , B ∗ Z = ωδˆ Ψ , B Z, (27) i δJ δφA h¯ i δJ where the second equality is a result of integration by parts in (26), and ωˆ is a nilpotent operator (which is also found in the Ward identity ωˆ Z = 0), namely, " −→ !# −→ m h¯ δ δ 2 ωˆ = JA + Lmσ,A , B ∗ , ωˆ = 0. (28) i δJ δφA

In terms of a generating functional Γ (B, φ, Σ, φ∗) of vertex Green’s functions with composite ﬁelds (on a background Bµ) given by a double Legendre transformation [94],

∗ ∗ A m m Γ (B, φ, Σ, φ ) = W (B, J, L, φ ) − JAφ − Lm [σ (φ, B) + Σ ] , W = (h¯ /i) ln Z, (29) where −→ −→ −→ ! ←− ←− δ W δ W δ W δ δ A = m = − m − = + m ( ) − = φ , Σ σ , B , JA Γ A Lmσ,A φ, B , Lm Γ m . δJA δLm δJ δφ δΣ Symmetry 2020, 12, 1985 8 of 26

∗ The variation δΨΓ (B, φ, Σ, φ ) induced by (27), with account taken of δΨΓ = δΨW, reads ←− −→ ←− −→ −→ −→ δ δ Γ δ δ Γ δ Γ δ = hh ii − hh ii + m ( ˆ ) − m ( ) hh ii δΨΓ δΨ A ∗ δΨ ∗ A σ,A φ, B σ,A φ, B ∗ m δΨ δφ δφA δφA δφ δφA δΣ " ←− −→ ! ) −→ " ←− −→ ! ) −→ ! i δ δ Γ δ δ δ Γ δ − m ( ˆ ) a + m ( ˆ ) n( ) hh ii Γ m σ,C φ, B ∗ , Φ a Γ m σ,C φ, B ∗ , σ φ, B n δΨ h¯ δΣ δφC δΦ δΣ δφC δΣ " ←− −→ ! ) −→ i n D δ δ Γ δ + (− )e(σ )+e(φ ) n ( ) m ( ˆ ) D hh ii 1 σ,D φ, B Γ m σ,C φ, B ∗ , φ n δΨ h¯ δΣ δφC δΣ " ←− ) −→ −→ ! −→ m D A δ Aa δ δ Γ δ hhδΨii +(− )e(σ )+e(φ )e(φ ) m ( ) n ( ˆ ) 00−1 1 σ,D φ, B , Γ n σ,A φ, B G a ∗ m , (30) δΣ δΦ δφD δΣ

δΨΓ ≡ ωˆ ΓhhδΨii, hhδΨii ≡ δΨ(φˆ, B), (31) where −→ Aa δ φˆ A = φA + ih¯ G00−1 , Φa = φA, Σm , δΦa −→ ←− ←− ←− ! 00 δ δ δ n δ G = Fb, Fa = Γ − Γ σ (φ, B), Γ . ab δΦa δφA δΣn ,A δΣm

−i/¯hW i/¯hW In (30), (31), ωˆ Γ is a Legendre transform of ωˆ W ≡ e ωˆ e , with ωˆ given by (28), 2 2 and therefore ωˆ Γ inherits the nilpotency of ωˆ , namely, ωˆ Γ = ωˆ W = 0. From (30) it follows, ∗ according to [95,96], that the generating functional Γ (B, φ, Σ, φ ) is gauge-independent, δΨΓ = 0, on the extremals δΓ δΓ = = 0, (32) δφA δΣm so that the effective action Γeff = Γeff (B, Σ) with composite and background ﬁelds deﬁned as

∗ Γeff (B, Σ) = Γ (B, φ, Σ, φ )|φ=φ∗=0 (33)

A ∗ is gauge-independent, δΨΓeff = 0, on the extremals (32) restricted to the hypersurface φ = φA = 0 of A ∗ vanishing quantum fields φ and antifields φA. This result on gauge dependence, in fact, holds true for general gauge theories, whose gauge algebra may be open, and whose gauge generators may be reducible, given an appropriate field-antifield structure. Besides, since the restricted generating functional Z (B, J, L), as we set φ∗ = 0 in (24), satisfies the identity (23), which is also valid for the related functional W (B, J, L) as we substitute Z = exp [(i/¯h)W], the effective action Γeff (B, Σ) defined by (29), (33) obeys an equality resulting from a Legendre transform of (23) written in terms of ∗ A W (B, J, L) = W (B, J, L, φ )|φ∗=0 and then reduced to φ = 0, ( −→ −→ ) Z h pq i δ p ···rˆ···p δ dDx D (B) ξq + g f {p}rqˆ Σ 1 k ξq Γ (B, Σ) = 0, (34) µ p µ1···µl p1···pk eff δBµ δΣµ1···µl as a consequence of the following identity implied by the notation f {p}rqˆ in (13), due to the antisymmetry of the structure constants: −→ −→ p ···p δ p ···rˆ···p δ f {p}rqˆ Σ 1 k = − f {p}rqˆ Σ 1 k . (35) µ1···µl p ···rˆ···p µ1···µl p1···pk 1 k δΣ ··· δΣµ1···µl µ1 µl

The effective action is thereby invariant, δξ Γeff = 0, under the local transformations

p = pq ( ) q p1···pk = {p}rqˆ p1···rˆ···pk q δξ Bµ Dµ B ξ , δξ Σµ1···µl g f Σµ1···µl ξ , (36) Symmetry 2020, 12, 1985 9 of 26

p which consist of the initial gauge transformations for the background ﬁeld Bµ and of the local SU(N) p1···pi transformations for the ﬁelds Σµ1···µj . Note that we assume the existence of a “deep” gauge-invariant regularization preserving the Ward identities (see, e.g., [14]), and expect the corresponding renormalized generating functionals to obey the same properties as the unrenormalized ones.

2.2. Background Field Interpretation In order to interpret the generating functional Z (B, J, L) in (21), note that the modified Faddeev–Popov action SFP (φ, B) defined as (18)–(20) is invariant under the finite local transformations

V −1 V −1 −1 −1 Aµ, b, c¯, c → V Aµ, b, c¯, c V , Bµ → VBµV + g V∂µV (37) and acquires the form ←− SFP (φ, B) = S0 (A + B) + Ψ (φ, B) s q , (38) where

S (A + B) = S (A)| = S (A)| , 0 0 [∂µ,•]→[Dµ(B),•] 0 Dµ(A)→Dµ(A+B) ←− ←− Ψ (φ, B) = Ψ (φ)| , s q = s , (39) [∂µ,•]→[Dµ(B),•] Dµ(A)→Dµ(A+B) ←− and the variation s q reads explicitly ←− Aµ, b, c¯, c s q = Dµ (A + B) , c , 0, b, g/2 [c, c]+ , p p p p ←− pq q p pqr q r Aµ, b , c¯ , c s q = Dµ (A + B) c , 0, b , g/2 f c c . (40)

Thus, the Landau and Feynman gauges (12) are modiﬁed to the respective background gauges

p pq q|µ p p pq q|µ χL (φ, B) = Dµ (B) A , χF (φ, B) = b + Dµ (B) A . (41)

In the background ﬁeld method, a quantum action SFP (φ, B) given by (38), (40) is known as the Faddeev–Popov action with a background ﬁeld Bµ. The quantum action SFP (φ, B) is invariant under A ←− global transformations of φ , with a nilpotent generator s q and an anticommuting parameter λ:

A A←− δλSFP (φ, B) = 0, δφ = φ s qλ, e (λ) = 1. (42)

Infinitesimally, the local transformations (37) for the fields Aµ, Bµ are known as background transformations, δ (Aµ, Bµ), and the transformations of Aµ, Bµ corresponding to the modified Slavnov ←− b variation s q in (38), (40), (42) are known as quantum transformations, δq(Aµ, Bµ),

p p p p δb Aµ = g Aµ, T ξ , δbBµ = Dµ (B) , T ξ , p p δq Aµ = Dµ (A + B) , T ξ , δqBµ = 0, while the classical action S0 (A + B) is invariant under both types of such transformations. In this regard, the family of background gauges χp (φ, B) = χ˜ p (A, B) + (α/2) bp, parameterized by α 6= 0 and deﬁned as (39), χp (A, B) = χp (A)| , [∂µ,•]→[Dµ(B),•] Symmetry 2020, 12, 1985 10 of 26 with the Nakanishi–Lautrup ﬁelds bp integrated out of (21) by the shift bp → bp + α−1χ˜ at the vanishing sources, J = L = 0, transforms the vacuum functional Z (B) to the representation (for the convenience of Section3, we denote A ≡ Q),

Z i h i Z(B) = dQ dc dc exp S (Q + B) + S (Q, B) + S (Q, B; c, c) , h¯ 0 gf gh 1 Z Z S (Q, B) = − dDx χ˜ pχ˜ p, S (Q, B) = dDx c¯p δ χ˜ p , (43) gf 2α gh q ξ→c where the gauge-fixing term Sgf = Sgf (Q, B) is invariant, δbSgf = 0 under the background p prq r q transformations, due to δbχ˜ = g f χ˜ ξ , which can be employed as a definition for the quantum action in background gauges χ˜ p (Q, B) depending on the quantum and background fields, with the related background and quantum transformations (see also [5]),

p pq q p prq r q δbBµ = Dµ (B) ξ , δbQµ = g f Qµξ , p p pq q (44) δqBµ = 0, δqQµ = Dµ (Q + B) ξ .

The quantum action and the integrand of Z(B) in (43) are invariant under the residual local transformations (37),

V −1 V −1 −1 −1 Qµ, c¯, c → V Qµ, c¯, c V , Bµ → VBµV + g V∂µV , (45) corresponding, at the inﬁnitesimal level δξ Bµ, Qµ, c¯, c , to the background transformations δb along with some compensating local transformations of the ghost ﬁelds:

p p p p p p prq r q prq r q δξ Bµ, Qµ, c¯ , c = (δbBµ, δbQµ, g f c¯ ξ , g f c ξ ).

In view of the above, we interpret Z (B, J, L) in (6), (18), (19), (21) as a generating functional of Green’s functions for Yang–Mills theories with composite fields in the background field method, or as a generating functional of Green’s functions with composite and background fields in such theories. As has been shown, this interpretation provides for such theories the existence of a corresponding gauge-invariant (36) effective action, being gauge-independent on the extremals (32) for the extended generating functional of vertex functions (29) when these extremals are restricted to the hypersurface of vanishing antifields and quantum fields.

3. Volovich–Katanaev Model Let us examine the model of two-dimensional gravity with dynamical torsion described in terms i of a zweibein eµ and a Lorentz connection ωµ by the action [50]

Z 1 1 S (e, ω) = d2x e R ijRµν − T iTµν − γ , (46) 0 16α µν ij 8β µν i where α, β, γ are constant parameters, and the following notation is used:

i e = det eµ, ij ij Rµν = ε Rµν, Rµν = ∂µων − (µ ↔ ν), (47) i i ij Tµν = ∂µeν + ε ωµeνj − (µ ↔ ν).

Here, the indices of quantities transforming under the local Lorentz group are denoted by Latin characters, i, j, k ... (i = 0, 1), with εij being a constant antisymmetric second-rank pseudo-tensor subject to the normalization ε01 = 1. The indices of quantities transforming as (pseudo-)tensors under the general coordinate transformations are labelled by Greek characters, λ, µ, ν ... (λ = 0, 1). The Latin Symmetry 2020, 12, 1985 11 of 26

indices are raised and lowered by the Minkowski metric ηij (+, −), and the Greek indices, by the i j metric tensor gµν = ηijeµeν. The action (46) is invariant under the local Lorentz transformations, i 0i 0 eµ → eµ, ωµ → ω µ,

0i i eµ = (Λeµ) , 0 i −1 i −1 i i ik (Ω µ)j = (ΛΩµΛ )j + (Λ∂µΛ )j, (Ωµ)j ≡ ε ηkjωµ, (48) or, inﬁnitesimally, with a parameter ζ,

i ij δζ eµ = ε eµjζ, δζ ωµ = −∂µζ, (49)

0 0 as well as under the general coordinate transformations, x → x = x (x),

λ i 0i 0 ∂x i e → e (x ) = 0 e (x), µ µ ∂x µ λ λ 0 0 ∂x ωµ → ω (x ) = 0 ωλ(x), (50) µ ∂x µ implying the inﬁnitesimal ﬁeld variations, with some parameters ξµ,

i i ν i ν ν ν δξ eµ = eν∂µξ + (∂νeµ)ξ , δξ ωµ = ων∂µξ + (∂νωµ)ξ . (51)

The given model can be quantized using the Faddeev–Popov method, since the gauge transformations (49), (51) form a closed algebra:

[δζ(1), δζ(2)] = 0,

[δξ(1), δξ(2)] = δξ(1,2), (52) [ ] = 0 δζ, δξ δζ , where µ ν µ µ ν 0 µ ξ (1,2) = ξ (1)∂νξ (2) − (∂νξ (1))ξ (2), ζ = (∂µζ)ξ . In [75], a quantum theory for the gauge model (46), (49), (51), (52) has been presented according to the background field method, by using an ansatz for the vacuum functional which corresponds to Z (B) of (43) for a Yang–Mills theory with the Nakanishi–Lautrup fields removes using a background gauge, see also [5]. Thus, the initial classical fields are ascribed the sets of quantum Q and background B fields, i i i to be denoted by Q = (qµ, qµ) and B = (eµ, ωµ), with the expressions for gµν, as well as e, (Ωµ)j in (47), (48), being related only to the background fields. One also associates the gauge transformations (49), (51) with two forms of infinitesimal transformations, background δb and quantum δq, being constructed by analogy with (44), so that the action S0(Q + B) in (46) remains invariant under both of these forms of transformations,

i ij i ν i ν i ij i ν a ν δbeµ = ε eµjζ + eν∂µξ + (∂νeµ)ξ , δbqµ = ε qµjζ + qν∂µξ + (∂νqµ)ξ , ν ν ν ν (53) δbωµ = −∂µζ + ων∂µξ + (∂νωµ)ξ , δbqµ = qν∂µξ + (∂νqµ)ξ ,

i i ij i i ν i i ν δqeµ = 0, δqqµ = ε (eµj + qµj)ζ + (eν + qν)∂µξ + (∂νeµ + ∂νqµ)ξ , ν ν (54) δqωµ = 0, δqqµ = −∂µζ + (ων + qν)∂µξ + (∂νωµ + ∂νqµ)ξ .

One next introduces the Faddeev–Popov ghosts (c, c), (cµ, cµ) according to the respective number of gauge parameters ζ, ξµ in (49), (51),

e(c, cµ, c, cµ) = 1, gh (c, cµ) = −gh(c, cµ) = 1, Symmetry 2020, 12, 1985 12 of 26 and considers an analogue [5] of the generating functional of Green’s functions, (c¯, c¯µ, c, cµ) = (C, C),

Z i h i Z(B, J) = dQ dC dC exp S (Q + B) + S (Q, B) + S (Q, B; C, C) + JQ , (55) h¯ 0 gf gh

µ µ i where J = (Ji , J ) are sources to the quantum ﬁelds Q = (qµ, qµ), and the functional Sgf = Sgf(Q, B) is constructed using some background gauge functions χ, χµ (for the respective gauge parameters µ ζ, ξ ) as one demands its invariance, δbSgf = 0, under the background transformations, whereas the ghost term Sgh = Sgh(Q, B; C, C) is then determined by Z = 2 ( + µ ) Sgh d x cδqχ c δqχµ (ζ,ξµ)→(c,cµ) . (56)

Let us choose the gauge functions χ = χ(Q, B) and χµ = χµ(Q, B) to be linear in the quantum i ﬁelds Q = (qµ, qµ) as in [75],

µν λν i µλ µ χ = eg ∇µqν , χµ = eg eµi∇λqν , g gλν = δν , (57)

i i j where e = det eµ, gµν = ηijeµeν, and ∇µ is a covariant derivative acting on an arbitrary (psedo-)tensor ν1...νl j1...jn i ik λ ﬁeld T , in terms of the connection (Ωµ) = ε η ωµ and the Christoffel symbols Γ µ1...µk i1...im j kj µν

1 Γλ = gλσ(∂ g + ∂ g − ∂ g ), (58) µν 2 ν µσ µ νσ σ µν by the rule

ˆ ∇ ν1...νl j1...jn = ν1...νl j1...jn − λˆ ν1...νl j1...jn + {ν} ν1...λ...νl j1...jn µTµ1...µk ∂µTµ1...µk Γ { }T ˆ Γ ˆ Tµ1...µk i1...im i1...im µ µ µ1...λ...µk i1...im µλ i1...im

{j} ν1...νl j1...pˆ...jn pˆ ν1...νl j1...jn + (Ωµ) T − (Ωµ) T , (59) pˆ µ1...µk i1...im {i} µ1...µk i1...pˆ...im with the shorthand notation, pk ∈ {p1,..., pm}, νk ∈ {ν1,..., νn},

pˆ µ ···µ p µ ···µ {µ} µ ···νˆ···µ µ µ ···ν ···µ F T 1 n = F k T 1 n , G T 1 n = G k T 1 k n , {i} i1···pˆ···im ∑ ik i1···pk···im νˆ i1···im ∑ νk i1···im pk νk

{i} i1···pˆ···im ik i1···pk···im νˆ i1···im νk i1···im F T ··· = F T ··· , G T = G T ··· ··· , (60) pˆ µ1 µn ∑ pk µ1 µn {µ} µ1···νˆ···µn ∑ µk µ1 νk µn pk νk so that the covariant derivative ∇µ features the standard properties (with F, G being arbitrary (psedo-)tensor ﬁelds)

µν ∇σgµν = ∇σg = 0, ∇µ(FG) = F∇µG + (∇µF)G. (61)

The above objects make it possible to construct the gauge-ﬁxing term Sgf as a functional quadratic in the background gauges χ, χµ (including some numeric parameters a, b)

1 Z S = d2x e−1 aχ2 + bχ χµ (62) gf 2 µ and invariant under the local Lorentz transformations

0i i 0i i eµ = (Λeµ) , qµ = (Λqµ) , 0 i −1 i −1 i 0 (Ω µ)b = (ΛΩµΛ )j + (Λ∂µΛ )j, qµ = qµ, (63) Symmetry 2020, 12, 1985 13 of 26

0 0 as well as under the general coordinate transformations, x → x = x (x),

λ λ 0i 0 ∂x i 0 0 ∂x e (x ) = 0 e (x), ω (x ) = 0 ωλ(x), µ ∂x µ λ µ ∂x µ λ λ 0i 0 ∂x i 0 0 ∂x q (x ) = 0 q (x), q (x ) = 0 qλ(x). (64) µ ∂x µ λ µ ∂x µ The field transformations (63) and (64) coincide infinitesimally with the background transformations (53), which thereby implies the fulfillment of δbSgf = 0. Having this in mind, as well as the fact that the non-zero quantum transformations (54), with account taken of (59), can be represented as

i ij i i ν i i ν ij ν δqqµ = ε (eµj + qµj)c + (eν + qν)∇µc + (∇νeµ + ∇νqµ)c − ε ων(eµj + qµb)c , ν ν δqqµ = −∇µc + (ων + qν)∇µc + (∇νωµ + ∇νqµ)c

µ µ under ζ → c, ξ → c , the ghost contribution Sgh in (56) reads Z 2 µ µ ν ν Sgh = d x e −c∇µ∇ c + c∇ [(∇νωµ + ∇νqµ)c + (ων + qν)∇µc ] ij µ ν λ +ε c eµi∇ [(eνj + qνj)(c − ωλc )] µ ν i i λ i i λ o + c eµi∇ [(∇λeν + ∇λqν)c + (eλ + qλ)∇νc ] . (65)

The quantum action in (55) represented by (46), (57), (62), (65) turns out to be invariant (which is also valid for the integrand in (55) once the sources are put to zero J = 0 under the assumption ( ) = ( ) = δ x ∂µδ x x=0 0) with respect to the background transformations (53) accompanied by a set of compensating local transformations for the ghost ﬁelds [75],

µ µ ν µ µ ν δc = (∂µc)ξ , δc = −c ∂νξ + (∂νc )ξ , µ µ µ ν µ µ ν (66) δc = −c ∂µζ + (∂µc)ξ , δc = −c ∂νξ + (∂νc )ξ .

From (53) and (66), it follows that the generating functional Z(B, J) in (55) is invariant [75] under i the initial gauge transformations (49), (51) of the background ﬁelds B = (eµ, ωµ) along with the µ µ following local transformations of the sources J = (Ji , J ):

µ k µ ν µ µ ν µ ν µ µ ν i ik δJi = −εi Jk ζ − Ji ∂νξ + ∂ν(Ji ξ ), δJ = −J ∂νξ + ∂ν(J ξ ), εj ≡ ε ηkj , (67) which implies the property Z 2 µ µ ν i ν µ δ (JQ) = d x ∂µF , F (x) ≡ Ji qν + J qν ξ for the source term JQ in (55) and extends an inﬁnitesimal tensor transformation law for the sources µ µ µ ν µ ν Ji and J by adding the contributions Ji ∂νξ and J ∂νξ , indeed,

µ h k µ ν µ µ νi µ ν µ ν µ µ ν µ ν δJi = −εi Jk ζ − Ji ∂νξ + (∂ν Ji )ξ + Ji ∂νξ , δJ = [−J ∂νξ + (∂ν J )ξ ] + J ∂νξ .

Given the invariance of Z(B, J) = exp{(i/h) W(B, J)} under (49), (51), (67), one establishes the invariance [75] of a functional Γ = Γ(B, Q) deﬁned as

δW δΓ Γ(B, Q) = W(B, J) − JQ, Q = , J = − , Q = (qi , q ) (68) δJ δQ µ µ Symmetry 2020, 12, 1985 14 of 26 under the background transformations (53) of the ﬁelds B and Q,

Z δΓ δΓ δ Γ = d2x δ B (x) + δ Q (x) = 0, b δB (x) b δQ (x) b which entails that the effective action Γeff(B) of the background ﬁeld method,

Γeff(B) = Γ(B, Q)|Q=0 , proves to be invariant, δ(ζ,ξ)Γeff = 0, under the gauge transformations (49), (51) of the background i ﬁelds B = (eµ, ωµ).

Composite Field Introduction Let us extend the generating functional (55) proposed in [75] to a functional Z(B, J, L), as we m introduce a set of local background-dependent composite ﬁelds σ (Q, B) with sources Lm,

m i1···ik µ1···µl σ (Q, B) = σ ··· (Q (x) , B (x)) , Lm = L (x) , m = (x, i ,..., i , µ ,..., µ ) , µ1 µl i1···ik 1 k 1 l and denote the entire quantum action by S(Q, B; C, C),

Z i Z(B, J, L) = dQ dC dC exp S(Q, B; C, C) + JQ + L σm (Q, B) . (69) h¯ m

In this setting, we demand that the extended functional Z(B, J, L) inherit the local symmetry i of Z(B, J) under the transformations (49), (51), (67) of the background fields B = (eµ, ωµ) and the = ( µ µ) i1···im ( ) = sources J Ji , J . To do so, we impose the condition that the composite fields σµ1···µn x i1···im ( ) ( ) σµ1···µn (Q x , B x ) should behave as tensors with respect to the Lorentz (63) and general coordinate (64) transformations of the quantum and background fields,

0 i1···im i1 im j1···jm σ ··· (x) = Λ ··· Λ σ ··· (x) , µ1 µn j1 jm µ1 µn 0 0 ν1 νn 0 0 i1···im ∂x ∂x i1···im σµ ···µn (x ) = 0 ··· 0 σν ···νn (x) , x = x (x) . (70) 1 ∂x µ1 ∂x µn 1

i1···im Generally, a composite field σµ1···µn (Q, B) transforming as (70) is multiplicative with respect to i i ij i the quantum fields Q = (qµ, qµ) and the background field objects eµ, gµν, Rµν , Tµν , see (47). Such a field may also include some background covariant derivatives ∇µ acting as (59), (58) and having the i1···im 6= properties (61). Given this, any composite field limited by σµ1···µn (Q, 0) 0 is allowed to contain the λ i background fields B only via the covariant derivative ∇µ, defined in terms of Γµν, (Ωµ)j and acting on i qµ, qµ as follows: i i λ i i j λ ∇µqν = ∂µqν − Γµνqλ + (Ωµ)jqν , ∇µqν = ∂µqν − Γµνqλ .

i1···im The transformations (70) correspond inﬁnitesimally to local tensor variations δσµ1···µn with parameters ζ and ξµ,

i1···im {i} i1···pˆ···im i1···im νˆ i1···im ν δσ ··· = ε σ ··· ζ + σ ∂ ξ + (∂νσ ··· )ξ . (71) µ1 µn pˆ µ1 µn µ1···νˆ···µn {µ} µ1 µn

Under these assumptions and the invariance of the vacuum functional in (69) with respect to the background transformations (53), accompanied by the compensating local transformations (66) of the ghost ﬁelds, the modiﬁed generating functional Z(B, J, L) in (69) is invariant under the initial gauge Symmetry 2020, 12, 1985 15 of 26

i transformations (49), (51) of the background ﬁelds B = (eµ, ωµ) along with the local transformations µ µ ···µ (67) of the sources J = J , Jµ and some local transformations of the sources L 1 n , i i1···im

µ1···µn pˆ µ1···µn µ1···νˆ···µn {µ} µ1···µn ν δL = −ε L ζ − L ∂ ˆξ + ∂ν(L ξ ) , (72) i1···im {i} i1···pˆ···im i1···im ν i1···im

µ1···µn ν which diverges from the inﬁnitesimal tensor transformation law by the contribution L ∂νξ , i1···im

µ1···µn h pˆ µ1···µn µ1···νˆ···µn {µ} µ1···µn νi µ1···µn ν δL = −ε L ζ − L ∂ ˆξ + (∂ν L )ξ + L ∂νξ , i1···im {i} i1···pˆ···im i1···im ν i1···im i1···im

m and provides for the source term Lmσ in (69) the transformation property Z m 2 µ µ ν1···νn i1···im µ δ (Lmσ ) = d x ∂µG , G ≡ L σ ··· ξ . i1···im ν1 νn

The invariance of Z(B, J, L) under (49), (51), (67), (72) can be represented as ( Z h i δ δ 2 ij + i ( ν) + ( i ) ν + − + ( ν) + ( ) ν d x ε eµjζ eν ∂µξ ∂νeµ ξ i ∂µζ ων ∂µξ ∂νωµ ξ δeµ δωµ h µ µ i δ δ − εk J ζ + Jν(∂ ξµ) − ∂ (J ξν) − [Jν(∂ ξµ) − ∂ (Jµξν)] i k i ν ν i µ ν ν δJµ δJi ) pˆ µ1···µn µ1···νˆ···µn {µ} µ1···µn ν δ − [ε L ζ + L (∂νˆξ ) − ∂ν(L ξ )] µ ···µ Z(B, J, L) = 0. (73) {i} i1···pˆ···im i1···im i1···im δL 1 n i1···im

Let us introduce a functional Γ = Γ(B, Q, Σ) given by the double Legendre transformation

m m Γ (B, Q, Σ) = W (B, J, L) − JQ − Lm [σ (Q, B) + Σ ] , W = (h¯ /i) ln Z (74)

m = i1···im in terms of additional ﬁelds Σ Σµ1···µn (x),

δW δW δW δΓ δσm δΓ = m = − m − = + − = Q , Σ σ , B , J Lm , Lm m . δJ δLm δJ δQ δQ δΣ

The effective action Γeff (B, Σ) with composite and background ﬁelds,

Γeff (B, Σ) = Γ (B, Q, Σ)|Q=0 , (75) then satisﬁes the identity R 2 h ij i ν i νi δ ν ν δ d x ε eµjζ + eν(∂µξ ) + (∂νeµ)ξ i + −∂µζ + ων(∂µξ ) + (∂νωµ)ξ δeµ δωµ " #) pˆ ··· ··· ··· + ε Σi1 im ζ δ + Σi1 im (∂ ξ{µ}) δ + (∂ Σi1 im )ξν δ Γ (B, Σ) = 0, {i} µ1···µn i1···pˆ···im µ1···µn νˆ i1···im ν µ1···µn i1···im eff δΣµ ···µ δΣ δΣµ ···µ 1 n µ1···νˆ···µn 1 n due to integration by parts in (73), as written in terms of Γ(B, Q, Σ), which is completed by setting m Q = σ (0, B) = 0. Using the latter identity for Γeff (B, Σ) and the following consequences, cf. (35), of the notation (60),

pˆ i1···im δ {i} i1···pˆ···im δ ε Σµ ···µn = ε Σµ ···µn , {i} 1 i1···pˆ···im pˆ 1 i1···im δΣµ1···µn δΣµ1···µn

i1···im {µ} δ i1···im νˆ δ Σµ ···µn (∂νˆξ ) = Σ (∂{µ}ξ ) , 1 i1···im µ1···νˆ···µn i1···im δΣ δΣ ··· µ1···νˆ···µn µ1 µn Symmetry 2020, 12, 1985 16 of 26

j ji k 0 1 0 1 by virtue of εk = ε ηik = εj (ε1 = ε0 = −1, ε0 = ε1 = 0), we find that Γeff (B, Σ) is invariant, δ(ζ,ξ)Γeff = 0, under a set of local transformations comprised by the gauge transformations (49), (51) of i the background fields B = (eµ, ωµ) along with the infinitesimal local tensor transformations

i1···im {i} i1···pˆ···im i1···im νˆ i1···im ν δ Σ ··· = ε Σ ··· ζ + Σ ∂ ξ + (∂νΣ ··· )ξ (76) (ζ,ξ) µ1 µn pˆ µ1 µn µ1···νˆ···µn {µ} µ1 µn

i1···im of the additional ﬁelds Σµ1···µn , cf. (70), (71).

4. Gribov–Zwanziger Model Let us examine the Gribov–Zwanziger model [23,24] implementing the concept of Gribov horizon [22] in quantum Yang–Mills theories by using a non-local composite ﬁeld. To this end, p µ p we consider a Euclidean form of the Faddeev–Popov action SFP (φ) in Landau gauge χL (φ) = ∂ Aµ, (8), (9), (10), and examine a non-local horizon functional H (A),

Z Z pq D D prt r −1 qst s|µ 2 H (A) = γ d x d y f gAµ (x) K (x; y) f gA (y) + D N − 1 , (77)

µ where, in Euclidean metric Aµ = A , we use the signature ηµν = (−, +,..., +) under the Wick rotation 0 0 p|0 p|0 µ x → ix , A → iA , SFP → iSFP and maintain the summation convention AµBµ = AµB ; besides, K−1 is the inverse,

Z pr Z rq dDz K−1 (x; z)(K)rq (z; y) = dDz (K)pr (x; z) K−1 (z; y) = δpqδ (x − y) , (78)

µ p of the Faddeev–Popov operator K in terms of the gauge condition ∂ Aµ = 0,

pq pq 2 prq r µ pq qp K (x; y) = δ ∂ + g f Aµ∂ δ (x − y) , K (x; y) = K (y; x) , (79) while γ is a Gribov thermodynamic parameter [23,24], introduced in a self-consistent way by using a gap equation (horizon condition) for a Gribov–Zwanziger action SGZ = SGZ (φ),

∂E Z vac = 0, exp −h¯ −1E ≡ dφ exp −h¯ −1S , ∂γ vac GZ where Evac is the vacuum energy, and the action SGZ reads

SGZ (φ) = SFP (φ) − H (A) . (80)

A generating functional of Green’s functions ZH (J) for the quantum theory under study can be given in terms of a Faddeev–Popov action shifted by a constant value, SFP (φ) − H (0), Z n −1 h A io ZH (J) = ZH (J, L)|L=1 , ZH (J, L) = dφ exp −h¯ SFP (φ) − H (0) + JAφ + Lσ (A) , (81) where L = L (x), e (L) = gh (L) = 0, is a source to the non-local composite ﬁeld Z D trp r ˜ −1 pq qst s|µ σ (A)(x) = γ d y f gAµ (x) (K ) (x; y) f gA (y) , (82) with K˜ −1 being the inverse, Z Z dDz (K˜ −1)pr (x; z) (K˜)rq (z; y) = dDz (K˜)pr (x; z) (K˜ −1)rq (z; y) = δpqδ (x − y) , (83) Symmetry 2020, 12, 1985 17 of 26 of an operator K˜ deﬁned for a quantity Fp = Fp (x), Z D pq q µ p p p p p d y (K˜) (x; y) F (y) ≡ ∂µ, [D (A) , F] (x) , Aµ (x) = T Aµ (x) , F (x) = T F (x) , (84) which implies pq µ pq K˜ (x; y) = ∂ Dµ (A) δ (x − y) , (85) and reduces to the operator K of (79) under the Landau gauge condition due to SFP (φ) in the path integral (81).

4.1. Formal Background Introduction

Let us now formally extend the generating functional ZH (J, L) with a non-local composite field (81)–(84) to the case of an additional background field Bµ with a covariant derivative Dµ (B) and the gauge properties (16), (17), by using the approach (2), (19), as adapted to Euclidean QFT, which implies a modification of derivatives ∂µ → Dµ (B) in (81), according to Z n h io Z (B, J, L) = Z (J, L)| = dφ exp −h¯ −1 S (φ, B) − H (0) + J φA + Lσ (A, B) , H H ∂µ→Dµ(B) FP A (86) p where SFP (φ, B) is the Faddeev–Popov action in the Landau background gauge χL (φ, B) = 0, see (41), and σ (A, B) is a non-local composite field on a background, Z D trp r ˜ −1 pq qst s|µ σ (A, B)(x) = γ d y f gAµ (x) (KB ) (x; y) f gA (y) . (87)

˜ −1 ˜ −1 ˜ Here, KB is a modiﬁed operator K as in (78), with the related inverse KB given by the replacement K˜ → K˜ B, Z D ˜ pq q µ p p p d y (K)B (x; y) F (y) ≡ Dµ (B) , [D (A + B) , F] (x) , F (x) = F (x) T , (88) and having the explicit form

˜ pq pr rq|µ (K)B (x; y) = Dµ (B) D (A + B) δ (x − y) . (89)

In the particular case, cf. (81),

ZH (B, J) = ZH (B, J, L)|L=1 , we obtain the generating functional Z n −1 h Aio ZH (B, J) = dφ exp −h¯ SGZ (φ, B) + JAφ , SGZ (φ, B) ≡ SFP (φ, B) − H (A, B) , (90) with a non-local term H (A, B) given by

Z Z pq D D prt r −1 qst s|µ 2 H (A, B) = γ d x d y f gAµ (x) KB (x; y) f gA (y) + D N − 1 . (91)

−1 ˜ Here, KB is the inverse of an operator KB as in (83), which is identical with KB in (88) being pq q|µ expressed, due to SFP (φ, B) in (90), by utilizing the gauge condition Dµ (B) A = 0 and the properties of f pqr including the Jacobi identity,

h 2 µ µ 2 µi KB (x; y) = ∂ + g ∂µB + g Aµ + 2Bµ ∂ + g Aµ + Bµ B δ (x − y) , (92) Symmetry 2020, 12, 1985 18 of 26

pq pq prq r r where Aµ, Bµ are matrices with the elements Aµ , Bµ = f Aµ, Bµ , and KB (x; y) is related to

K˜ B (x; y) in (89) as follows:

µ µ KB (x; y) = K˜ B (x; y) − g Dµ (B) , A δ (x − y) = D (A + B) Dµ (B) δ (x − y) . (93)

The operator KB extends the original operator K in (79) and exhibits the properties

pq qp KB|B=0 = K, (KB) (x; y) = (KB) (y; x) , (94) which can be veriﬁed by a direct calculation: Z D pq qp q prq h rs s|µi q d y (KB) (x; y) − (KB) (y; x) F (y) = g f Dµ (B) A F (x) = 0.

By virtue of (94), one may formally interpret SGZ (φ, B) as a Gribov–Zwanziger action on a background Bµ, with the corresponding non-local horizon term H (A, B) in (91), (93) and the generating functional ZH (B, J) in (90). However, since the background introduction (2), (19) has been applied to a non-local composite field it is natural to examine the consistency of the above interpretation with the symmetries exhibited by ZH (B, J). To this end, it is useful to recast ZH (B, J) in a local form by extending the configuration space according to [97], in such a way that one introduces a set of pq pq pq pq pq pq commuting (ϕ¯µ , ϕµ ) and anticommuting (ω¯ µ , ωµ ) auxiliary fields, where ϕ¯µ and ϕµ are mutually complex-conjugate,

pq pq pq pq pq pq pq pq e ϕ¯µ , ϕµ , ω¯ µ , ωµ = (0, 0, 1, 1) , gh ϕ¯µ , ϕµ , ω¯ µ , ωµ = (0, 0, −1, 1) .

This makes it possible to construct the parametrization Z n −1 o h −1 i exp h¯ [H (A, B) − H (0, B)] = dϕ¯ dϕ dω¯ dω exp −h¯ Sγ (A, B; ϕ¯ ϕ, ω¯ , ω) , (95) where Z D h rp pq rq|µ rp pq rq|µ 1/2 prq r|µ pq pqi Sγ = d x −ϕ¯µ KB ϕ + ω¯ µ KB ω + iγ g f A ϕ¯µ + ϕµ , (96) as we denote Z Z pq rq D pq rq pq rq D pq rq KB ϕµ (x) = d y KB (x, y) ϕµ (y) , KB ωµ (x) = d y KB (x, y) ωµ (y) .

In the conﬁguration space Φ = (φ, ϕ¯, ϕ, ω¯ , ω), the functional ZH (B, J) of (90) takes the form Z n −1 h Aio ZH (B, J) = dΦ exp −h¯ SGZ (Φ, B) + JAφ , (97) where the local action SGZ (Φ, B) of the Gribov–Zwanziger theory on a background reads (note the antisymmetry of f pqr) Z 1/2 D µ T SGZ (Φ, B) = SFP (φ, B) − H (0, B) + SK (Φ, B) − iγ g d x Tr A ϕ¯µ − ϕµ , Z D D h µ T µ T i SK (Φ, B) ≡ d x d y Tr −ϕ¯ (x) KB (x, y) ϕµ (y) + ω¯ (x) KB (x, y) ωµ (y) . (98)

pqr Using the explicit form of KB (x, y) in (93), the antisymmetry of f , and a repeated integration by parts, one can remove the delta-function δ (x − y) absorbed in KB (x, y) and present SK (Φ, B) in (98) as follows: Z D h µ ν T µ ν Ti SK (Φ, B) = d x Tr −ϕ¯ Dν (A + B) D (B) ϕµ + ω¯ Dν (A + B) D (B) ωµ . (99) Symmetry 2020, 12, 1985 19 of 26

The action SGZ (Φ, B) is invariant under the global SU (N) transformations

T T U T T −1 Aµ, Bµ, b, c¯, c, ϕ¯µ, ϕµ, ω¯ µ, ωµ → U Aµ, Bµ, b, c¯, c, ϕ¯µ, ϕµ, ω¯ µ, ωµ U . (100)

The inﬁnitesimal form of (100)

pq pq prs rq s qrs pr s pq T T δςF = g f F ς + g f F ς , F = Aµ, Bµ, b, c¯, c, ϕ¯µ, ϕµ, ω¯ µ, ωµ (101) produces a unit Jacobian (due to the antisymmetry of f pqr) in the integration measure of (97) and leaves the functional ZH (B, J) invariant under inﬁnitesimal global SU (N) transformations for the background ﬁeld Bµ and the sources JA, having the adjoint representation form

pq pq prs rq s qrs pr s pq δςG = g f G ς + g f G ς , G = Bµ, Jµ(A), J(b), J(c¯), J(c) . (102)

This behavior of ZH (B, J) obviously entails an invariance of the restricted generating functional ZH (J) under the global SU (N) transformations of the sources; however, the global SU (N) invariance of ZH (J) fails to translate itself into a local symmetry of the background functional ZH (J, B). Indeed, let us apply an inﬁnitesimal form of the local change of variables (U → V)

V 0 V 0 −1 −1 −1 Φ → Φ , Bµ → Bµ = VBµV + g V ∂µV , (103) with a unitary matrix V = V (ξ), ξ = ξ (x), to the integrand in (97), which produces a unit Jacobian −1 2 −1 and a variation δξ SGZ = δξ SK in (98), so that the presence of extra derivatives ∂µV and ∂ V in the transformed expression Z 0 0 D h µ ν T −1 µ ν T −1i SK Φ , B = d x Tr −Vϕ¯ Dν (A + B) D (B) ϕµV + Vω¯ Dν (A + B) D (B) ωµV Z D h −1 µ ν T −1 µ ν Ti 6= d x Tr −V Vϕ¯ Dν (A + B) D (B) ϕµ + V Vω¯ Dν (A + B) D (B) ωµ ,

0 0 implies SK (Φ , B ) 6= SK (Φ, B), also involving a local non-invariance of the background horizon term, 0 0 A H (A , B ) 6= H (A, B). Finally, in view of δξ JAφ = 0, the functional ZH (B, J) is not invariant under the background gauge transformations of Bµ with the local SU (N) transformations of JA, since the latter do not compensate the variation δξ SGZ (Φ, B) 6= 0.

4.2. Modiﬁed Background Introduction and Effective Action

The local non-invariance of ZH (B, J) can be explained by the fact that the background field Bµ has been introduced directly into the non-local horizon term H (A) by means of (93), while localizing the resultant background term H (A, B) by the auxiliary fields (ϕ¯µ, ϕµ) and (ω¯ µ, ωµ) does not provide them with a covariant derivative as in (19), which is evident from (99). In order to remedy this issue, we examine an alternative introduction of a background field, i.e., by using a local parametrization of the original horizon term H (A) at a point before a background has been switched on. To this end, let us consider the expressions (95)–(98) when restricted to Bµ = 0 and recast the functional ZH (J) given by (81) in the form Z n −1 h Aio ZH (J) = dΦ exp −h¯ SGZ (Φ) + JAφ , SGZ (Φ) = SGZ (Φ, 0) , Φ = (φ, ϕ¯, ϕ, ω¯ , ω) . Symmetry 2020, 12, 1985 20 of 26

For a consideration of the auxiliary ﬁelds (ϕ¯, ω¯ ) and ϕT, ωT on equal terms, note that the action µ SGZ (Φ) in the above integrand with the Landau gauge condition ∂µ A = 0 encoded in exp [−hS¯ FP (φ)] is equivalent to an action SGZ (Φ) which arises from replacing K (x, y) by K (x, y) deﬁned as

1 K (x, y) ≡ K (x, y) + K˜ (x, y) , K˜ (x, y) = K (x, y) + g ∂ , Aµ δ (x − y) . (104) 2 µ Using integration by parts, permutations under the sign of Tr, and the Landau gauge condition in the path integral, one brings the equivalent action SGZ (Φ) to the form Z 1/2 D µ T SGZ (Φ) = SFP (φ) − H (0) + SK (Φ) − iγ g d x Tr A ϕ¯µ − ϕµ , (105) Z 1 D n ν µ T ν µ h Tio SK (Φ) ≡ d x Tr [D (A) , ϕ¯ ] ∂ ϕ + (∂ ϕ¯ ) D (A) , ϕ − (ϕ¯, ϕ → ω¯ , ω) , 2 ν µ ν µ which invites a natural introduction of a background, SGZ (Φ) → SGZ (Φ, B), according to (19), Z 1/2 D µ T SGZ (Φ, B) = SFP (φ, B) − H (0) + SK (Φ, B) − iγ g d x Tr A ϕ¯µ − ϕµ , (106) Z 1 D n ν µ h Ti ν µ h Tio SK (Φ, B) = d x Tr [D (A + B) , ϕ¯ ] Dν (B) , ϕ + [D (B) , ϕ¯ ] Dν (A + B) , ϕ 2 µ µ Z D pq pq|rs rs|µ pq pq|rs rs|µ − (ϕ¯, ϕ → ω¯ , ω) ≡ d x (−ϕ¯µ KB ϕ + ω¯ µ KB ω ).

Let us consider the expression

1 Z hF, Gi ≡ − dDx Tr D (A + B) , F [Dµ (B) , G] + D (B) , F [Dµ (A + B) , G] , e (F) = e (G) , 2 µ µ and present it in the form Z Z D pq pq|rs rs D D pq pq|rs rs hF, Gi = d x F KB G = d x d y F (x) KB (x; y) G (y) .

Then, due to the (anti)symmetry of hF, Gi under the replacement F ↔ G, we ﬁnd

pq|rs rs|pq KB (x; y) = KB (y; x) .

Given the above, we interpret SGZ (Φ, B) in (106) as an alternative local Gribov–Zwanziger action on the background Bµ, with the corresponding modiﬁed background horizon term H (A, B) deﬁned as Z n −1 o h −1 i exp h¯ [H (A, B) − H (0, B)] = dϕ¯ dϕ dω¯ dω exp −h¯ Sγ (Φ, B) , H (0, B) ≡ H (0) , (107) where Z 1/2 D µ T Sγ (Φ, B) = SK (Φ, B) − iγ g d x Tr A ϕ¯µ − ϕµ Z D h pq pq|rs rs|µ pq pq|rs rs|µ 1/2 prq r|µ pq pqi = d x −ϕ¯µ KB ϕ + ω¯ µ KB ω + iγ g f A ϕ¯µ + ϕµ .

By construction, the action SGZ (Φ, B) in (106) is invariant under the local transformations (103), which implies a unit Jacobian in the inﬁnitesimal case and leads to an invariance of the background generating functional Z n −1 h Aio ZH (B, J) = dΦ exp −h¯ SGZ (Φ, B) + JAΦ , ZH (0, J) = ZH (J) , (108) Symmetry 2020, 12, 1985 21 of 26 with respect to local transformations of the sources and the background ﬁeld:

p pq q p prq r q δξ Bµ = Dµ (B) ξ , δξ (Jµ(A), J(b), J(c¯), J(c)) = g f (Jµ(A), J(b), J(c¯), J(c)) ξ . (109)

The transformations (103) also provide a local invariance of the modiﬁed background horizon term H (A, B) in (107),

p pq q p prq r q δξ H (A, B) = 0, δξ Bµ = Dµ (B) ξ , δξ Aµ = g f Aµξ .

Introducing the generating functionals of connected WH (B, J) and vertex ΓH (B, φ) Green’s functions on a background, −→ ←− δ δ = − −1 ( ) = ( ) − A A = = − ZH exp h¯ WH , ΓH B, φ WH B, J JAφ , φ WH, JA ΓH A , (110) δJA δφ one translates the invariance of ZH (B, J) under (109) into an invariance of ΓH (B, φ) under the following local transformations, cf. (22), (23), (34), (36),

p pq q p prq r q δξ Bµ = Dµ (B) ξ , δξ (Aµ, b, c¯, c) = g f (Aµ, b, c¯, c) ξ , (111) which consist of the gauge transformations for the background field Bµ and of the local SU(N) A transformations for the quantum fields φ , so that the background effective action Γeff (B) for the Gribov–Zwanziger model defined as

Γeff (B) = ΓH (B, φ)|φ=0 (112) is invariant, δξ Γeff = 0, under the gauge transformations of the background ﬁeld Bµ.

5. Conclusions In the present article, we have approached the issue of a joint introduction of composite and background fields into non-Abelian quantum gauge models on the basis of symmetries exhibited by the generating functional of Green’s functions. Thus, we examine the Yang–Mills theory, the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion, and the Gribov–Zwanziger theory. Our systematic analysis of the problem focuses on quantum Yang–Mills theories and local composite fields. As our first approach, we consider a theory quantized according to Faddeev and Popov and including some composite fields (1), with the resultant generating functional Z (J, L) having an inherent symmetry under global SU (N) transformations of the sources JA, Lm for the quantum φA and composite σm (φ) fields. As our second approach, we examine a theory quantized in the background field method (3), with the background generating functional Z (B, J) being inherently invariant under local SU (N) transformations of the sources JA accompanied by gauge transformations of the background field Bµ with an associated covariant derivative Dµ (B). In the first approach, we introduce a background field Bµ equipped with Dµ (B), whereas in the second approach we m introduce some composite fields σ (φ, B) with sources Lm. In doing so, we demand that the B L functional Z (B, J, L) given by Z (J, L) → Z (B, J, L) and Z (B, J) → Z (B, J, L) reflect the symmetries inherent in Z (J, L) and Z (B, J). The resultant Z (B, J, L) is thereby invariant under the local SU (N) transformations of JA, Lm combined with the gauge transformations of Bµ, whereas our first and second approaches are equivalent in the sense of (2) and (4), (5) with ∂µ → Dµ (B) given by (19). In quantum Yang–Mills theories with composite and background fields, we have introduced an ∗ extended (due to antifields φA) generating functional of vertex Green’s functions (24), (29), including a background effective action (33), and investigated its properties. For the generating functional of Symmetry 2020, 12, 1985 22 of 26 vertex Green’s functions (effective action), gauge dependence has been obtained in terms of a nilpotent operator with composite and background fields (30), (31), and conditions of on-shell independence from gauge-fixing have been established. Thus, the generating functional of vertex functions is found to be gauge-independent on its extremals (32), whereas the background effective action is is independent of any specific choice of gauge-fixing at the extremals (32) restricted to the hypersurface of vanishing ∗ A antifields φA and quantum fields φ . The background effective action Γeff (B, Σ), depending on the m m background field Bµ and a set of auxiliary fields Σ associated with the composite fields σ , is found m to be invariant under the gauge transformations of Bµ and the local SU(N) transformations of Σ , see (36). L Introduction of composite fields in the approach Z (B, J) → Z (B, J, L) starting from a background generating functional Z (B, J) has been extended beyond the Yang–Mills case by considering the Volovich–Katanaev model [50] of two-dimensional gravity with dynamical torsion (in terms of a i zweibein eµ and a Lorentz connection ωµ) quantized using the background field method in [75] and exhibiting a gauge-invariant background effective action. Thus, the quantum theory [75] has i1···im µ1···µn been modified by introducing local composite fields σ ··· with sources L , as we demand for µ1 µn i1···im i1···im σµ1···µn to behave as tensors (70) under the Lorentz (63) and general coordinate (64) transformations i i of background B = (eµ, ωµ) and quantum Q = (qµ, qµ) fields. These transformations coincide infinitesimally with the background transformations δbB and δbQ in (53), constructed by analogy with the Yang–Mills case (44). The corresponding background effective action Γeff (B, Σ) defined as (69), (74), (75) has been found invariant under the gauge transformations (49), (51) of background fields B i1···im i1···im combined with the local transformations (76) of additional fields Σµ1···µn related to σµ1···µn . As in the Yang–Mills case, (76) is an infinitesimal form of tensor transformations, cf. (70), (71). B Introduction of background fields in the approach Z (J, L) → Z (B, J, L) starting from a generating functional Z (J, L) has been extended beyond the case of local composite fields by considering the Gribov–Zwanziger model [23,24] implementing the Gribov horizon [22] for quantum Yang–Mills theories in Landau gauge by means of an additive horizon functional H (A) which can be presented in terms of a non-local composite field σ (A). Direct introduction of a background field Bµ into the composite field, σ (A) → σ (A, B), produces a formally consistent (94) background-modified generating functional for the Gribov–Zwanziger theory, see ZH (B, J) in (90) with the corresponding non-local horizon term H (A, B) in (91), (93). However, it has been shown that ZH (B, J) does not inherit (in a localized form) the symmetry of the original Gribov–Zwanziger functional ZH (J) in (81) under global SU (N) transformations of the sources, as H (A, B) fails to be invariant under local SU (N) transformations of quantum Yang–Mills field Aµ combined with gauge transformations of the background field Bµ. Further, representing the original horizon term H (A) in a local parametrization using auxiliary fields according to [97], we bring the related functional ZH (J) to an equivalent (under the Landau gauge) form with a modified horizon term H (A). This local parametrization admits a natural introduction of a covariant derivative Dµ (B) as acting on the auxiliary fields by extension ∂µ → Dµ (B) according to (19), which implies a background horizon term H (A, B) invariant under the local SU (N) transformations of Aµ along with the gauge transformations of Bµ. The resulting modified background functional ZH (B, J) in (108) is consistent, ZH (B, J)|B=0 = ZH (J), and invariant under the local SU (N) transformations of the sources combined with the gauge transformations of the background field. The corresponding background effective action Γeff (B) for the Gribov–Zwanziger model defined as (106), (108), (110), (112) proves to be invariant under the gauge transformations δξ Bµ = Dµ (B) , ξ of the background field Bµ. This effective action seems advantageous as a starting point in approaching a renormalization analysis of the Gribov–Zwanziger model as one accounts for the horizon in the background field method. One should note, the calculations maybe independently done in the Computer Algebra Systems, like Mathematica Singular or Maple. We checked the correctness some of the equations with use of the Mathematica. Symmetry 2020, 12, 1985 23 of 26

One more application of the suggested approach may be developed for QCD-gauge theory of strong interactions with SU(3) gauge group [33,88]-to describe hadron (meson and baryon. e.g., proton, m ¯ α neutron) particles as the composite fields, σ (u¯, u, d, d, Aµ), of up u and down d quarks (including left- α an right-handed spinors) of spin 1/2 and Aµ, α = 1, ..., 8 of spin 1 gluons fields. Here, the Grassmann parity e(σm) should equal to 1 for baryon and to 0 for meson particles. Another interesting application of the presented background field method with composite fields is a consideration of so-called Generalized Lagrange space (for metric fields) to exploit its properties of curvature, torsion and deflection in order to take into account the asymmetries and anisotropies arising from physical phenomena basically at the cosmological level.

Author Contributions: Conceptualization, P.Yu.M. and A.A.R.; methodology, P.Yu.M. and A.A.R.; software, P.Yu.M., A.A.R.; validation, P.Yu.M., A.A.R.; formal analysis, P.Yu.M., A.A.R.; investigation, P.Yu.M., A.A.R.; resources, P.Yu.M., A.A.R.; data curation, P.Yu.M., A.A.R.; writing—original draft preparation, P.Yu.M.; writing—review and editing, P.Yu.M., A.A.R.; visualization, P.Yu.M., A.A.R.; supervision, P.Yu.M., A.A.R.; project administration, A.A.R.; funding acquisition, P.Yu.M., A.A.R. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: The work of P.Yu.M. has been supported by the Tomsk State University Competitiveness Improvement Program. A.A.R. is grateful for the support from the Program of Fundamental Research under the Russian Academy of Sciences, 2013–2020. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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